Dirichlet forms, quasiregular functions and Brownian motion (Q1100307)

In this article it is shown that given a quasiregular function \(\phi\) on an open set U in \({\mathbb{R}}^ n\) there exists a diffusion \(X_ t\) in U such that \(\phi\) maps \(X_ t\) into n-dimensional Brownian motion. The process is constructed from a Dirichlet form which can be described explicitly. This opens for the use of stochastic methods in the study of quasiregular mappings. It also raises interesting questions regarding the connection between the stochastic methods and methods from degenerate elliptic differential equations, the theory of \(A_ p\) weights and the nonlinear potential theory associated to quasiregular functions. Applications are given to boundary behaviour and value distribution.